My math book says two lines that are each perpendicular to the same
third line are not necessarily parallel to each other. How can that
be? They would not touch, and isn't that the definition of parallel
lines?

At 1:45, the angle between the hands of a clock is 142.5 degrees. When
is the next time the angle between the hands will be 142.5 degrees? In
addition to that specific problem, this talks about general strategies
for solving problems involving angles between the hands of a clock.

A geometry teacher wonders if his student has proven that a quadrilateral with one
pair of congruent sides and one set of congruent angles is a parallelogram. By
following the steps from another Dr. Math conversation cited by the teacher, Doctor
Peterson illustrates the proof's hidden assumption with a counter-example.

Considering the four symmetry transformations--reflection, rotation,
translation, and glide reflection--is it possible to express
transformations in the two-dimensional plane as a composition of at
most three reflections?